Let A be the set of integers and let n be a fixed positive integer. Define a relation on A be saying xRy if n divides x - y or x + y.

Just giving an example, if n = 5 we know 3R7 since 5 divides 3 + 7 = 10. Also it works with 12R2 12 - 2 = 10. Prove R is an equivalence relation.

A relation is an equivalence relation if and only if it has these three properties:
1)Reflexive: xRx for every x in the set.
x- x= 0 so that is easy

2)Symmetric: if xRy then yRx.
If xRy then either x- y is a multiple of n or x+ y is a multiple of n. For the first of those, you need the fact that y- x= -(x- y).

3)Transitive: if xRy and yRz then xRz.
If xRy then either x- y is a multiple of n or x+ y is a multiple of n. If yRz then either y- z is a multiple of n or y+ z is a multiple of n. You really need to look at 4 cases:
a) x- y is a multiple of n and y- z is a multiple of n. What is (x-y)+ (y- z)?
b) x- y is a multiple of n and y+ z is a multiple of n. What is (x-y)+ (y+ z)?
c) x+ y is a multiple of n and y- z is a multiple of n. What is (x+y)- (y- z)?
d) x+ y is a multiple of n and y+ z is a multiple of n. What is (x+y)- (y+z)?